The Absolute Derivative In relativity we typically deal with two types of quantities: fields, which are defined everywhere, and particle properties, which are defined only along a curve or world line. The familiar covariant derivative is appropriate when we need to differentiate a field. A field is a function of all four coordinates, and the…
The Schwarzschild Metric A Lagrangian that can be used to describe geodesics is [itex]F = g_{\mu\nu}v^\mu v^\mu[/itex], where [itex]v^\mu = dx^\mu/ds[/itex] is the four-velocity. In the equatorial plane of the Schwarzschild metric this is $$F = (1 – 2m/r)^{-1} (dr/ds)^2 + r^2(d\phi/ds)^2 – (1 – 2m/r)(dt/ds)^2$$ The canonical momenta are [itex]p_\mu = \partial F/\partial v^\nu…
Tetrads and Notation A spacetime is often described in terms of a tetrad field, that is, by giving a set of basis vectors at each point. Let the vectors of the tetrad be denoted by eaμ, where μ is a tensor index and where a is a tetrad index that serves to number the vectors…